# Talk:Multiscale modeling

This article is a short review on multiscale modeling. Multiscale modeling refers to a class of models, where models at different scales in space and/or time are used in order to better describe the system under study. In my opinion, it should be distinguished from:

• multiscale analytical derivations, which consist, starting from one model at a given scale, to obtain other models at coarser scales using reduction or coarse-graining techniques. Examples include two-scale expansions in homogenization, Chapman-Enskog expansion techniques to derive transport equations, etc...
• multiscale algorithms, which use, as a numerical tool, coarse-grained descriptions in order to accelerate the computation at the original fine scale. Examples include multi-grid techniques, fast multipole methods, the fast-Fourier transform, various non-linear approximation techniques, etc...

In this paper, all these aspects of multiscale techniques are treated as multiscale modeling.

It is of course impossible to review all the aspects of multiscale modelling in a few pages. Examples of multiscale algorithms which are not mentioned include multiscale finite element methods, stabilization methods based on a multiscale interpretation for finite element computations (subgrid scale method), Large Eddy Simulation techniques for turbulence modelling, multiscale image analysis, etc... Examples of applications where multiscale modelling becomes today a fundamental tool include the modelling of the human cardio-vascular system, the modelling of the vehicle traffic flow, climate modelling, pattern formation for magnetic materials, etc...

Let me mention a few points which, in my opinion, are not sufficiently stressed.

One fundamental aspect of multiscale modeling is the mathematical analysis of the obtained coupled system, to prove its well-posedness, and in particular the consistency of the models used at various scales. The coupling of a model at one scale to a model at another scale requires care, in particular in situation where a fine model is used in some parts of the domain, and a coarser elsewhere. For example, in the quasi-continuum method, the coupling of an atomistic model with a model from continuum mechanics cannot be done if the continuum mechanics model is in some sense, a limit of the atomistic model. Such consistency checks between the models are one of the major ingredient of multiscale modeling, a classical example being the mathematical studies aiming at bridging the gap from molecular dynamics to Boltzmann equations, to Navier Stokes or Euler equations. Behind multiscale modelling is hidden the fundamental question of deriving the law at the macroscopic scale, from the law at the microscopic scale, which is a Holy Grail of mathematical physics. This includes, for example, the justification of the Fourier laws and the computation of the transport coefficients (heat conductivity, viscosity of a Newtonian fluid, etc) from models at the microscopic scale, which is a currently very active research field. Roughly speaking, the situation is rather well understood for minimization problems (which would correspond to zero temperature models in mechanics), but much more complicated for dynamical systems, especially when they are out of equilibrium.

Concerning more practical aspects, one major ingredient which is often missing since it is generally very delicate to obtain, is an a priori and, even better, a posteriori, error estimate of the error introduced by the use of a coarse model, instead of the fine model. A posteriori error estimates are in particular required in order to perform model adaptation, namely using the fine model only where it is needed. When only one model is used, mesh adaptation is typically one example where such a posteriori estimates are used: based on local a posteriori error estimate, the mesh is automatically refined where needed, for a given prescribed accuracy on an output. Deriving error estimates for coarse-graining and reduction techniques is a major challenge of multiscale modelling.

Let me finally give a few minor comments on the text:

• On Figure 1, I personally do not consider this picture as a realistic one: in many cases, there is no clear separation of space and time scales between the various models. This is what makes multiscale modelling often very delicate. For example, for polymeric fluids, the time scale for the kinetic models is of the same order as the time scale for the macroscopic equations of continuum mechanics (the ratio of timescales being one of the major nondimensional number in this field, called the Weissenberg number).
• The difference between the Car-Parrinello molecular dynamics, and the QM-MM methods is not clear. Both seem to be a coupling between Molecular Dynamics and Quantum Mechanics.
• Domain decomposition techniques were certainly proposed before the provided reference.
• On Figure 2, is it U which is reconstructed from u, or is it the other way around ?

## Reviewer B:

This is a very nice review paper on multiscale modeling. Here follows are certain comments.

1. The paper aims for the introduction of multiscale modeling. It also discusses the multiscale methods, multiscale algorithms and multiscale analysis. It maybe helpful to add several lines on the relations among them, or give definitions on such terminologies, otherwise, the people who firstly touch such topic may be quite confused.

2. In the domain decomposition method part, it may be better to mention that quasicontinuum method is also of the same spirit for the sake of the coherence of the paper.

3. A conclusion part is missing unless the authors believe it is unnecessary.

4. It is more conventional to use quasicontinuum instead of quasi-continuum.

5. The references should be checked. In particular, the styles for  and  are different though both were published in Comm. Math. Sci..

## Author Jianfeng : Reply to reviewers

We thank the reviewers for these helpful suggestions and comments. Please find our replies below following the referee report.

Reviewer A: This article is a short review on multiscale modeling. Multiscale modeling refers to a class of models, where models at different scales in space and/or time are used in order to better describe the system under study. In my opinion, it should be distinguished from:
• multiscale analytical derivations, which consist, starting from one model at a given scale, to obtain other models at coarser scales using reduction or coarse-graining techniques. Examples include two-scale expansions in homogenization, Chapman-Enskog expansion techniques to derive transport equations, etc...
• multiscale algorithms, which use, as a numerical tool, coarse-grained descriptions in order to accelerate the computation at the original fine scale. Examples include multi-grid techniques, fast multipole methods, the fast-Fourier transform, various non-linear approximation techniques, etc...
In this paper, all these aspects of multiscale techniques are treated as multiscale modeling.

We took a broader perspective, since all these components are intimately related. As the reviewer observes below, one fundamental aspect of multiscale modeling is the mathematical analysis of these models. For that purpose, we need to understand the multiscale analysis tools that allow us to obtain macroscale models from microscale ones. In addition, the kind of multiscale algorithms that have been developed to accelerate the computation of original fine scale models are also intimately related to the more narrowly defined multiscale modeling techniques. A good example is the multi-grid method. Traditional multi-grid method was developed as a technique for solving detailed algebraic equations. But more recent developments allow the use of multi-physics models.

It is of course impossible to review all the aspects of multiscale modelling in a few pages. Examples of multiscale algorithms which are not mentioned include multiscale finite element methods, stabilization methods based on a multiscale interpretation for finite element computations (subgrid scale method), Large Eddy Simulation techniques for turbulence modelling, multiscale image analysis, etc... Examples of applications where multiscale modelling becomes today a fundamental tool include the modelling of the human cardio-vascular system, the modelling of the vehicle traffic flow, climate modelling, pattern formation for magnetic materials, etc...

Yes indeed the list goes on and on. We did not have space to review these more specific algorithms. We had to limit ourselves to the more general ideas.

In the revised version, we added a paragraph to briefly mention some of these more specific algorithms.

Let me mention a few points which, in my opinion, are not sufficiently stressed. One fundamental aspect of multiscale modeling is the mathematical analysis of the obtained coupled system, to prove its well-posedness, and in particular the consistency of the models used at various scales. The coupling of a model at one scale to a model at another scale requires care, in particular in situation where a fine model is used in some parts of the domain, and a coarser elsewhere. For example, in the quasi-continuum method, the coupling of an atomistic model with a model from continuum mechanics cannot be done if the continuum mechanics model is in some sense, a limit of the atomistic model. Such consistency checks between the models are one of the major ingredient of multiscale modeling, a classical example being the mathematical studies aiming at bridging the gap from molecular dynamics to Boltzmann equations, to Navier Stokes or Euler equations. Behind multiscale modelling is hidden the fundamental question of deriving the law at the macroscopic scale, from the law at the microscopic scale, which is a Holy Grail of mathematical physics. This includes, for example, the justification of the Fourier laws and the computation of the transport coefficients (heat conductivity, viscosity of a Newtonian fluid, etc) from models at the microscopic scale, which is a currently very active research field. Roughly speaking, the situation is rather well understood for minimization problems (which would correspond to zero temperature models in mechanics), but much more complicated for dynamical systems, especially when they are out of equilibrium.

This is a good point. We added a section: Challenges in multiscale modeling to highlight this issue.

Concerning more practical aspects, one major ingredient which is often missing since it is generally very delicate to obtain, is an a priori and, even better, a posteriori, error estimate of the error introduced by the use of a coarse model, instead of the fine model. A posteriori error estimates are in particular required in order to perform model adaptation, namely using the fine model only where it is needed. When only one model is used, mesh adaptation is typically one example where such a posteriori estimates are used: based on local a posteriori error estimate, the mesh is automatically refined where needed, for a given prescribed accuracy on an output. Deriving error estimates for coarse-graining and reduction techniques is a major challenge of multiscale modelling.

Let me finally give a few minor comments on the text: On Figure 1, I personally do not consider this picture as a realistic one: in many cases, there is no clear separation of space and time scales between the various models. This is what makes multiscale modelling often very delicate. For example, for polymeric fluids, the time scale for the kinetic models is of the same order as the time scale for the macroscopic equations of continuum mechanics (the ratio of timescales being one of the major nondimensional number in this field, called the Weissenberg number).

True. But the picture is only an illustration. It is not meant to be taken literally.

The difference between the Car-Parrinello molecular dynamics, and the QM-MM methods is not clear. Both seem to be a coupling between Molecular Dynamics and Quantum Mechanics.

Car-Parrinello is used to solve type B problem, namely supply the inter-atomic forces in a molecular dynamics simulation, by resorting to quantum mechanics models. QM-MM is for type A problems. For example, when only a small part of the system needs to be treated using quantum mechanics and the rest can be treated using classical mechanics.

Domain decomposition techniques were certainly proposed before the provided reference.

Yes, that is true. So are most of the other references. We did not intend to list the original references, since it is very hard to do in some cases, such as in the case of domain decomposition.

On Figure 2, is it U which is reconstructed from u, or is it the other way around ?

Yes, thanks for spotting this big mistake!! We fixed it.

Reviewer B: This is a very nice review paper on multiscale modeling. Here follows are certain comments. 1. The paper aims for the introduction of multiscale modeling. It also discusses the multiscale methods, multiscale algorithms and multiscale analysis. It maybe helpful to add several lines on the relations among them, or give definitions on such terminologies, otherwise, the people who firstly touch such topic may be quite confused.

Yes we did in the revised version.

2. In the domain decomposition method part, it may be better to mention that quasicontinuum method is also of the same spirit for the sake of the coherence of the paper.

Good suggestion.

3. A conclusion part is missing unless the authors believe it is unnecessary.

Yes, good idea. We added that.

4. It is more conventional to use quasicontinuum instead of quasi-continuum.

We fixed that too.

5. The references should be checked. In particular, the styles for  and  are different though both were published in Comm. Math. Sci.

We fixed that.

## Reviewer A:

Most of my comments have been taken into account, except maybe some remarks on the bibliography, which is not exhaustive on certain aspects.